Integrand size = 17, antiderivative size = 81 \[ \int \frac {1}{(1-x)^{13/3} (1+x)^{2/3}} \, dx=\frac {3 \sqrt [3]{1+x}}{20 (1-x)^{10/3}}+\frac {27 \sqrt [3]{1+x}}{280 (1-x)^{7/3}}+\frac {81 \sqrt [3]{1+x}}{1120 (1-x)^{4/3}}+\frac {243 \sqrt [3]{1+x}}{2240 \sqrt [3]{1-x}} \] Output:
3/20*(1+x)^(1/3)/(1-x)^(10/3)+27/280*(1+x)^(1/3)/(1-x)^(7/3)+81/1120*(1+x) ^(1/3)/(1-x)^(4/3)+243/2240*(1+x)^(1/3)/(1-x)^(1/3)
Time = 0.04 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.43 \[ \int \frac {1}{(1-x)^{13/3} (1+x)^{2/3}} \, dx=-\frac {3 \sqrt [3]{1+x} \left (-319+423 x-297 x^2+81 x^3\right )}{2240 (1-x)^{10/3}} \] Input:
Integrate[1/((1 - x)^(13/3)*(1 + x)^(2/3)),x]
Output:
(-3*(1 + x)^(1/3)*(-319 + 423*x - 297*x^2 + 81*x^3))/(2240*(1 - x)^(10/3))
Time = 0.16 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.12, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {55, 55, 55, 48}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {1}{(1-x)^{13/3} (x+1)^{2/3}} \, dx\) |
\(\Big \downarrow \) 55 |
\(\displaystyle \frac {9}{20} \int \frac {1}{(1-x)^{10/3} (x+1)^{2/3}}dx+\frac {3 \sqrt [3]{x+1}}{20 (1-x)^{10/3}}\) |
\(\Big \downarrow \) 55 |
\(\displaystyle \frac {9}{20} \left (\frac {3}{7} \int \frac {1}{(1-x)^{7/3} (x+1)^{2/3}}dx+\frac {3 \sqrt [3]{x+1}}{14 (1-x)^{7/3}}\right )+\frac {3 \sqrt [3]{x+1}}{20 (1-x)^{10/3}}\) |
\(\Big \downarrow \) 55 |
\(\displaystyle \frac {9}{20} \left (\frac {3}{7} \left (\frac {3}{8} \int \frac {1}{(1-x)^{4/3} (x+1)^{2/3}}dx+\frac {3 \sqrt [3]{x+1}}{8 (1-x)^{4/3}}\right )+\frac {3 \sqrt [3]{x+1}}{14 (1-x)^{7/3}}\right )+\frac {3 \sqrt [3]{x+1}}{20 (1-x)^{10/3}}\) |
\(\Big \downarrow \) 48 |
\(\displaystyle \frac {9}{20} \left (\frac {3}{7} \left (\frac {9 \sqrt [3]{x+1}}{16 \sqrt [3]{1-x}}+\frac {3 \sqrt [3]{x+1}}{8 (1-x)^{4/3}}\right )+\frac {3 \sqrt [3]{x+1}}{14 (1-x)^{7/3}}\right )+\frac {3 \sqrt [3]{x+1}}{20 (1-x)^{10/3}}\) |
Input:
Int[1/((1 - x)^(13/3)*(1 + x)^(2/3)),x]
Output:
(3*(1 + x)^(1/3))/(20*(1 - x)^(10/3)) + (9*((3*(1 + x)^(1/3))/(14*(1 - x)^ (7/3)) + (3*((3*(1 + x)^(1/3))/(8*(1 - x)^(4/3)) + (9*(1 + x)^(1/3))/(16*( 1 - x)^(1/3))))/7))/20
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp [(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{ a, b, c, d, m, n}, x] && EqQ[m + n + 2, 0] && NeQ[m, -1]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(S implify[m + n + 2]/((b*c - a*d)*(m + 1))) Int[(a + b*x)^Simplify[m + 1]*( c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && ILtQ[Simplify[m + n + 2], 0] && NeQ[m, -1] && !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[ c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (SumSimplerQ[m, 1] || !SumSimp lerQ[n, 1])
Time = 0.12 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.37
method | result | size |
gosper | \(-\frac {3 \left (1+x \right )^{\frac {1}{3}} \left (81 x^{3}-297 x^{2}+423 x -319\right )}{2240 \left (1-x \right )^{\frac {10}{3}}}\) | \(30\) |
orering | \(\frac {3 \left (1+x \right )^{\frac {1}{3}} \left (-1+x \right ) \left (81 x^{3}-297 x^{2}+423 x -319\right )}{2240 \left (1-x \right )^{\frac {13}{3}}}\) | \(33\) |
risch | \(\frac {3 \left (\left (1-x \right ) \left (1+x \right )^{2}\right )^{\frac {1}{3}} \left (81 x^{4}-216 x^{3}+126 x^{2}+104 x -319\right )}{2240 \left (1+x \right )^{\frac {2}{3}} \left (1-x \right )^{\frac {1}{3}} \left (-1+x \right )^{3} \left (-\left (-1+x \right ) \left (1+x \right )^{2}\right )^{\frac {1}{3}}}\) | \(65\) |
Input:
int(1/(1-x)^(13/3)/(1+x)^(2/3),x,method=_RETURNVERBOSE)
Output:
-3/2240*(1+x)^(1/3)/(1-x)^(10/3)*(81*x^3-297*x^2+423*x-319)
Time = 0.06 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.60 \[ \int \frac {1}{(1-x)^{13/3} (1+x)^{2/3}} \, dx=-\frac {3 \, {\left (81 \, x^{3} - 297 \, x^{2} + 423 \, x - 319\right )} {\left (x + 1\right )}^{\frac {1}{3}} {\left (-x + 1\right )}^{\frac {2}{3}}}{2240 \, {\left (x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 1\right )}} \] Input:
integrate(1/(1-x)^(13/3)/(1+x)^(2/3),x, algorithm="fricas")
Output:
-3/2240*(81*x^3 - 297*x^2 + 423*x - 319)*(x + 1)^(1/3)*(-x + 1)^(2/3)/(x^4 - 4*x^3 + 6*x^2 - 4*x + 1)
Result contains complex when optimal does not.
Time = 126.95 (sec) , antiderivative size = 984, normalized size of antiderivative = 12.15 \[ \int \frac {1}{(1-x)^{13/3} (1+x)^{2/3}} \, dx=\text {Too large to display} \] Input:
integrate(1/(1-x)**(13/3)/(1+x)**(2/3),x)
Output:
Piecewise((81*(x + 1)**3*exp(I*pi/3)*gamma(1/3)/(216*(-1 + 2/(x + 1))**(1/ 3)*(x + 1)**3*exp(I*pi/3)*gamma(13/3) - 1296*(-1 + 2/(x + 1))**(1/3)*(x + 1)**2*exp(I*pi/3)*gamma(13/3) + 2592*(-1 + 2/(x + 1))**(1/3)*(x + 1)*exp(I *pi/3)*gamma(13/3) - 1728*(-1 + 2/(x + 1))**(1/3)*exp(I*pi/3)*gamma(13/3)) - 540*(x + 1)**2*exp(I*pi/3)*gamma(1/3)/(216*(-1 + 2/(x + 1))**(1/3)*(x + 1)**3*exp(I*pi/3)*gamma(13/3) - 1296*(-1 + 2/(x + 1))**(1/3)*(x + 1)**2*e xp(I*pi/3)*gamma(13/3) + 2592*(-1 + 2/(x + 1))**(1/3)*(x + 1)*exp(I*pi/3)* gamma(13/3) - 1728*(-1 + 2/(x + 1))**(1/3)*exp(I*pi/3)*gamma(13/3)) + 1260 *(x + 1)*exp(I*pi/3)*gamma(1/3)/(216*(-1 + 2/(x + 1))**(1/3)*(x + 1)**3*ex p(I*pi/3)*gamma(13/3) - 1296*(-1 + 2/(x + 1))**(1/3)*(x + 1)**2*exp(I*pi/3 )*gamma(13/3) + 2592*(-1 + 2/(x + 1))**(1/3)*(x + 1)*exp(I*pi/3)*gamma(13/ 3) - 1728*(-1 + 2/(x + 1))**(1/3)*exp(I*pi/3)*gamma(13/3)) - 1120*exp(I*pi /3)*gamma(1/3)/(216*(-1 + 2/(x + 1))**(1/3)*(x + 1)**3*exp(I*pi/3)*gamma(1 3/3) - 1296*(-1 + 2/(x + 1))**(1/3)*(x + 1)**2*exp(I*pi/3)*gamma(13/3) + 2 592*(-1 + 2/(x + 1))**(1/3)*(x + 1)*exp(I*pi/3)*gamma(13/3) - 1728*(-1 + 2 /(x + 1))**(1/3)*exp(I*pi/3)*gamma(13/3)), 1/Abs(x + 1) > 1/2), (81*(x + 1 )**3*gamma(1/3)/(216*(1 - 2/(x + 1))**(1/3)*(x + 1)**3*exp(I*pi/3)*gamma(1 3/3) - 1296*(1 - 2/(x + 1))**(1/3)*(x + 1)**2*exp(I*pi/3)*gamma(13/3) + 25 92*(1 - 2/(x + 1))**(1/3)*(x + 1)*exp(I*pi/3)*gamma(13/3) - 1728*(1 - 2/(x + 1))**(1/3)*exp(I*pi/3)*gamma(13/3)) - 540*(x + 1)**2*gamma(1/3)/(216...
\[ \int \frac {1}{(1-x)^{13/3} (1+x)^{2/3}} \, dx=\int { \frac {1}{{\left (x + 1\right )}^{\frac {2}{3}} {\left (-x + 1\right )}^{\frac {13}{3}}} \,d x } \] Input:
integrate(1/(1-x)^(13/3)/(1+x)^(2/3),x, algorithm="maxima")
Output:
integrate(1/((x + 1)^(2/3)*(-x + 1)^(13/3)), x)
\[ \int \frac {1}{(1-x)^{13/3} (1+x)^{2/3}} \, dx=\int { \frac {1}{{\left (x + 1\right )}^{\frac {2}{3}} {\left (-x + 1\right )}^{\frac {13}{3}}} \,d x } \] Input:
integrate(1/(1-x)^(13/3)/(1+x)^(2/3),x, algorithm="giac")
Output:
integrate(1/((x + 1)^(2/3)*(-x + 1)^(13/3)), x)
Timed out. \[ \int \frac {1}{(1-x)^{13/3} (1+x)^{2/3}} \, dx=\int \frac {1}{{\left (1-x\right )}^{13/3}\,{\left (x+1\right )}^{2/3}} \,d x \] Input:
int(1/((1 - x)^(13/3)*(x + 1)^(2/3)),x)
Output:
int(1/((1 - x)^(13/3)*(x + 1)^(2/3)), x)
\[ \int \frac {1}{(1-x)^{13/3} (1+x)^{2/3}} \, dx=\int \frac {1}{\left (x +1\right )^{\frac {2}{3}} \left (1-x \right )^{\frac {1}{3}} x^{4}-4 \left (x +1\right )^{\frac {2}{3}} \left (1-x \right )^{\frac {1}{3}} x^{3}+6 \left (x +1\right )^{\frac {2}{3}} \left (1-x \right )^{\frac {1}{3}} x^{2}-4 \left (x +1\right )^{\frac {2}{3}} \left (1-x \right )^{\frac {1}{3}} x +\left (x +1\right )^{\frac {2}{3}} \left (1-x \right )^{\frac {1}{3}}}d x \] Input:
int(1/(1-x)^(13/3)/(1+x)^(2/3),x)
Output:
int(1/((x + 1)**(2/3)*( - x + 1)**(1/3)*x**4 - 4*(x + 1)**(2/3)*( - x + 1) **(1/3)*x**3 + 6*(x + 1)**(2/3)*( - x + 1)**(1/3)*x**2 - 4*(x + 1)**(2/3)* ( - x + 1)**(1/3)*x + (x + 1)**(2/3)*( - x + 1)**(1/3)),x)